Riley hobson bence · Problem 26.29

Numerical methods

Mathematics — Methods · tensors-christoffel-symbols-metric-tensor

Problem

We may define Christoffel symbols of the first kind by

\Gamma_{ijk} = g_{il}\Gamma^l_{jk}.

Show that these are given by

\Gamma_{kij} = \frac{1}{2}\left(\frac{\partial g_{ik}}{\partial u^j} + \frac{\partial g_{jk}}{\partial u^i} - \frac{\partial g_{ij}}{\partial u^k}\right).

By permuting indices, verify that

\frac{\partial g_{ij}}{\partial u^k} = \Gamma_{ijk} + \Gamma_{jik}.

Using the fact that

\Gamma^l_{jk} = \Gamma^l_{kj}

, show that

g_{ij\,;\,k} \equiv 0,

i.e. that the covariant derivative of the metric tensor is identically zero in all coordinate systems.